Referring now to FIG. 1, a system 10 includes multiple transceivers 12-1, 12-2, . . . , and 12-N (collectively transceivers 12). The transceivers 12-1, 12-2, . . . , and 12-N include transmitters 14-1, 14-2, . . . , and 14-N (collectively transmitters 14) and receivers 16-1, 16-2, . . . , and 16-N (collectively receivers 16) that transmit/receive signals over a communications channel, respectively. In wireless applications, the transmitters 14 and receivers 16 transmit/receive signals via an antenna and/or an array of antennas 20-1, 20-2, . . . , 20-N (collectively antennas 20). Signals output by the transmitters 14 travel through various paths (or multi-paths) having different lengths before arriving at the receivers 16.
Since multiple versions of the signal may arrive at different times and interfere with each other (commonly called inter symbol interference (ISI)), it becomes very difficult to extract the original data. In other words, when the communications system 10 transmits data at time intervals T and has a longest delay τmax with respect to the earliest path, a received symbol can be influenced by
      τ    max    Tprior symbols. To operate properly, the receiver 16 should compensate for the influence of ISI.
To reduce the number of prior symbols that can have impact on the current symbol, the original data stream may be multiplexed into N parallel data streams, each of which is modulated by a different frequency. The N parallel signals are transmitted. In effect, the time interval T is reduced by
      1    N    .Therefore, the number of prior symbols that influence the current symbol is reduced by
      1    N    ,which makes compensation for ISI easier.
Orthogonal frequency division multiplexing (OFDM) systems were developed to address multi-path, ISI and other problems. OFDM systems are used for high-speed communications through frequency selective channels. OFDM systems remove the inter-symbol interference (ISI) and are usually implemented using computationally efficient fast Fourier transform (FFT) techniques. Because of these advantages, OFDM is often used in wireless and wired communication systems such as wireless LAN (IEEE 802.11a and HIPERLAN/2), digital audio broadcasting (DAB), terrestrial digital video broadcasting (DVB-T), asymmetric digital subscriber line (ADSL), and very high-speed digital subscriber line (VDSL) systems.
Although OFDM has many advantages, OFDM is susceptible to carrier frequency offset (CFO), which may occur due to Doppler shift and/or from the mismatch between the oscillator frequencies of the transmitter 14 of one device and the receiver 16 of another device. The CFO attenuates the desired signal and introduces inter-carrier interference (ICI). As a result, the signal-to-noise ratio (SNR) decreases and consequently the performance of a OFDM system degrades. To overcome the adverse effects of the CFO, various carrier frequency synchronization methods have been developed. Since these synchronization methods cannot remove the CFO completely, the effect of a residual CFO must be accommodated.
In some systems, the SNR degradation due to the CFO is estimated for additive white Gaussian noise (AWGN) channels, time-invariant multipath channels and shadowed multipath channels. However, the SNR expressions are typically complex and only approximate the SNR. In other approaches, the effect of the CFO on the symbol error rate (SER) has been developed. However, the SER expression is complex and valid only for the AWGN channel.
Similar to single-carrier systems, differential demodulation in OFDM systems eliminates the need for channel estimation, which reduces the complexity of the receiver and training overhead as compared to coherent demodulation. However, these advantages are accompanied by approximately 3 dB degradation in signal-to-noise ratio (SNR).
Differential encoding can be done in the time-domain or the frequency-domain. The time-domain approach encodes a symbol differentially over two consecutive OFDM symbols at the same subcarrier. The frequency-domain approach encodes a symbol differentially over two adjacent subcarriers of the same OFDM symbol. Time-domain differential demodulation performs well when the channel changes slowly over time so that the channel phase of two consecutive OFDM symbols is about the same. On the other hand, frequency-domain differential demodulation is a good choice when the multipath spread is small compared to the length of one OFDM symbol so that the channel phase between two adjacent subcarriers in the same OFDM symbol is about the same.
Although there has been research relating to the effect of the multipath and Doppler spread on differential demodulation, little is known about the effect of the CFO. The receiver carrier frequency should be synchronized with that of the transmitter to avoid performance degradation. A differential demodulation receiver, however, does not need to track the transmitter carrier phase.